The origin of this problem lies in systems of hyperbolic partial differential equations with matrix coefficients. The eigenvalues are propagation speeds; whentwo speeds coincide, new phenomena arise.
The talk will deal with the identification of multiple eigenvalues. I will prove that if A, B, C are three real symmetric n x n matrices, i congruent 2 mod 4, then there are real numbers a, b, c, not all zero such that aA + bB + cC has a multiple eigenvalue.
In the second part I will show that the discriminant of real symmetric matrices can be written as a sum of squares of polynomials.